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 May 20th, 2017, 06:49 PM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2 Problems on Series 1. The largest Interval $I$ such that the series $\displaystyle \sum_{n=1}^{\infty} \frac{x^n}{\sqrt{n}}$ converges whenever $\displaystyle x \in I$ is equal to A)$[-1,1]$ B)$[-1,1)$ C)$(-1,1]$ D)$(-1,1)$ My answer : Options A or B 2. Let $\displaystyle \sum a_n$ be a convergent series. Let $b_n=a_{n+1} - a_n$ for all $\displaystyle n \in N$. Then A) $\displaystyle \sum b_n$ should also be convergent and $\displaystyle (b_n) \rightarrow 0$ as $\displaystyle n \rightarrow \infty.$ B) $\displaystyle \sum b_n$ need not be convergent but $\displaystyle (b_n) \rightarrow 0$ as $\displaystyle n \rightarrow \infty.$ C) $\displaystyle \sum b_n$ is convergent but $(b_n)$ need not tend to zero as $\displaystyle n \rightarrow \infty.$ D) None of the above statements is true. My answer : Option A 3. Which of the following series converge ? A) $\displaystyle \sum_{n=1}^{\infty} (\frac{\log n}{n^{1+2\epsilon}}$ B) $\displaystyle \sum_{n=1}^{\infty} (\frac{(\log n)^2}{n^{1+2\epsilon}}$ C) $\displaystyle \sum_{n=1}^{\infty} (\frac{n^2+1}{n^3+n})$ D) $\displaystyle \sum_{n=1}^{\infty} (1+\frac{1}{n})^n$ I checked Option C, it is convergent and Option D is not convergent. Kindly check if Option A or B are Convergent ? Help me to solve series with logarithms. Kindly check and let me know If I'm wrong. Thanks Last edited by skipjack; May 20th, 2017 at 09:43 PM. May 20th, 2017, 10:25 PM #2 Global Moderator   Joined: Dec 2006 Posts: 20,755 Thanks: 2137 1. (B) 2. (A) 3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent. May 20th, 2017, 10:41 PM   #3
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 Originally Posted by skipjack 1. (B) 2. (A) 3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
Sorry to confuse you. Can you please check the below series now ?

A) $\displaystyle \sum_{n=1}^{\infty} (\frac{\log n}{n^{1+2e}})$
B) $\displaystyle \sum_{n=1}^{\infty} (\frac{(\log n)^2}{n^{1+2e}})$ May 20th, 2017, 10:56 PM   #4
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 Originally Posted by skipjack 1. (B) 2. (A) 3. The parentheses don't balance in (A) and (B), and ϵ is undefined. (C) and (D) are divergent.
Can you suggest some videos or books on series where I can get details of how to check if a series is convergent or divergent and the range of n as well as for the values or a.

It's confusing for me to estimate a series converges or not.

Thanks  May 21st, 2017, 08:44 AM #5 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,664 Thanks: 2644 Math Focus: Mainly analysis and algebra C is precisely $\sum \frac1n$ and thus divergent. The terms of D tend to $e$ and not $0$, so that is divergent. Tags problems, sequences, series Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post brh27 Calculus 3 November 9th, 2012 02:09 PM Ter Algebra 4 June 11th, 2012 09:59 PM MathematicallyObtuse Algebra 2 February 15th, 2011 02:03 AM Balic Calculus 1 January 7th, 2010 05:51 AM dagitt Calculus 6 May 30th, 2009 01:52 PM

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